Question

Given that 𝑓(𝑥) = 3𝑥 2 − 4𝑥 + 7, use the definition of the derivative to find 𝑓 ′ (𝑥).

Answer 1

Given that $𝑓(𝑥) = 3𝑥^2 − 4𝑥 + 7$ , let us find $𝑓'(𝑥)$:

$f'(x)=\lim\limits_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}= \lim\limits_{\Delta x\to 0}\frac{3(x+\Delta x)^2-4(x+\Delta x)+7-(3x^3-4x+7)}{\Delta x}= \lim\limits_{\Delta x\to 0}\frac{3x^3+6x\Delta x+3(\Delta x)^2-4x-4\Delta x+7-3x^2+4x-7}{\Delta x}= \lim\limits_{\Delta x\to 0}\frac{6x\Delta x+3(\Delta x)^2-4\Delta x}{\Delta x}= \lim\limits_{\Delta x\to 0}(6x+3\Delta x-4)=6x-4.$